On a system of equations with primes
Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 399-413.

Étant donné un entier n3, soient u 1 ,...,u n des entiers 2 et premiers entre eux deux à deux, soit 𝒟 une famille de sous-ensembles propres et non vides de {1,...,n} qui contient un nombre “suffisant” des éléments, et soit ε une fonction 𝒟{±1}. Est-ce qu’il existe au moins un nombre premier q tel que q divise le nombre iI u i -ε(I) pour un certain I𝒟, mais q ne divise pas u 1 u n  ? Nous donnons une réponse positive à cette question dans le cas où les u i sont des puissances de nombres premiers et on impose certaines restrictions sur ε et 𝒟.

Nous utilisons ce résultat pour prouver que, si ε 0 {±1} et A est un ensemble de trois ou plusieurs nombres premiers qui contient les diviseurs premiers de tous les nombres pB p-ε 0 pour lesquels B est un sous-ensemble propre, fini et non vide de A, alors A contient tous les nombres premiers.

Given an integer n3, let u 1 ,...,u n be pairwise coprime integers 2, 𝒟 a family of nonempty proper subsets of {1,...,n} with “enough” elements, and ε a function 𝒟{±1}. Does there exist at least one prime q such that q divides iI u i -ε(I) for some I𝒟, but it does not divide u 1 u n ? We answer this question in the positive when the u i are prime powers and ε and 𝒟 are subjected to certain restrictions.

We use the result to prove that, if ε 0 {±1} and A is a set of three or more primes that contains all prime divisors of any number of the form pB p-ε 0 for which B is a finite nonempty proper subset of A, then A contains all the primes.

DOI : 10.5802/jtnb.873
Classification : 11A05, 11A41, 11A51, 11D61, 11D79, 11R27
Mots clés : Agoh-Giuga conjecture, cyclic congruences, prime factorization, Pillai’s equation, Znam’s problem.
Leonetti, Paolo 1 ; Tringali, Salvatore 2

1 Università Bocconi via Sarfatti 25 20100 Milan, Italy
2 Texas A&M University at Qatar PO Box 23874 Education City DOHA, 5825 QATAR
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Leonetti, Paolo; Tringali, Salvatore. On a system of equations with primes. Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 399-413. doi : 10.5802/jtnb.873. http://www.numdam.org/articles/10.5802/jtnb.873/

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